Truncated 6-cubes

In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube.

There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square FACES of the 6-cube. Vertices of the tritruncated 6-cube are located inside the cubic cells of the 6-cube. {{-}}

Truncated 6-cube

Alternate names

• Truncated hexeract (Acronym: tox) (Jonathan Bowers)

Construction and coordinates

The truncated 6-cube may be constructed by truncating the vertices of the 6-cube at $1/(\sqrt{2}+2)$ of the edge length. A regular 5-simplex replaces each original vertex.

The Cartesian coordinates of the vertices of a truncated 6-cube having edge length 2 are the permutations of:

$$\left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right)$$

Related polytopes

The truncated 6-cube, is fifth in a sequence of truncated hypercubes:

Bitruncated 6-cube

Alternate names

• Bitruncated hexeract (Acronym: botox) (Jonathan Bowers)

Construction and coordinates

The Cartesian coordinates of the vertices of a bitruncated 6-cube having edge length 2 are the permutations of:

(0, ±1, ±2, ±2, ±2, ±2)

Related polytopes

The bitruncated 6-cube is fourth in a sequence of bitruncated hypercubes:

Tritruncated 6-cube

Alternate names

• Tritruncated hexeract (Acronym: xog) (Jonathan Bowers)

Construction and coordinates

The Cartesian coordinates of the vertices of a tritruncated 6-cube having edge length 2 are the permutations of:

(0, 0, ±1, ±2, ±2, ±2)

Related polytopes

The table below contains a set of 63 uniform 6-polytopes generated from the B₆ Coxeter plane, including the regular 6-cube and 6-orthoplex.

References

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd edition, Dover, New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com,
    • (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.

• o3o3o3o3x4x - tox, o3o3o3x3x4o - botox, o3o3x3x3o4o - xog

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary